Tortoises All the Way Forward
Tortoises All the Way Forward
By Ian Stewart, Professor of Mathematics, University of Warwick, United Kingdom
The tortoise looked worried.
“How fast do you say you can run?”
“Ten cubits per second,” Achilles replied, doing fifty one-handed press-ups.
“How far is a cubit?”
“Twice as long as you,” said Achilles.
The tortoise paused, counting on his toenails. Eventually he said: “Hmm. I can manage a tenth of a cubit per second. If I really hurry.”
“That makes me a hundred times faster than you.”
The tortoise counted some more. “Ninety-nine times faster. A hundred times as fast.”
“Pedant.”
“In rational debate,” the tortoise said, “accuracy is a virtue.” A thought struck him. “How long is the race, anyway?”
“A thousand cubits.”
“Oh. That’s a lot.”
“Yes. Not that it matters much. If it was a tenth of that I’d still beat you.”
“It’s not a fair contest,” the tortoise protested, inspecting Achilles’s left sandal — all he could see of him.
Achilles thought about that. He was conceited, and highly competitive, but honour demanded a fair fight. “I’ll give you a head start.”
“How much?” the tortoise asked. “My head’s not very long, you know.”
“I didn’t mean—” Achilles began.
The tortoise, looking hopeful, interrupted. “How about a thousand cubits?”
“A hundred,” Achilles said, with a shake of his noble brow. “Can’t say fairer than that, mate. I mean, give me a break, OK?”
The tortoise looked sad. “I was rather hoping you’d give I me a break.”
“Look, torty-boy: I’m a professional. There are limits, you know.”
The tortoise nodded, a slow up-and-down motion of his head as it protruded from his shell. “Done!”
“Done?” Achilles was surprised. This was too easy. “Hang on. What’s the catch?”
“No catch,” said the tortoise.
“You’re not expecting me to get so far ahead I lie down for a nap, so you can trundle past?”
“No, that’s from another version of this story.”
“Right, then: we’re agreed. Head start of 100 cubits.”
“Excellent,” said the tortoise. “You do realise I’ll win.”
Achilles stared at him. “Don’t be silly. It will take me 100 seconds to reach the tape. By then you’ll have gone 10 cubits, so you’ll have reached only the 110-cubit mark. Still 890 cubits to go. I’ll leave you trailing in my dust.”
“That doesn’t prove you can catch me,” the tortoise pointed out. “It’s an existence proof, not a constructive one.”
Achilles gave a frustrated sigh. “Look, after t seconds I’ll be distance 10t along the course, and you’ll be 100+t/10. We’ll be level when those expressions are equal. So t = 10 10/99. By then we’ll both have reached the 101 1/99 cubit mark. Just over a tenth of the way to the finish line.”
The tortoise, deciding he had too few claws, scratched some algebra in the sand. “I agree it looks that way. But you have to catch me first. And I have it on good authority that you can’t.”
“Why ever not?”
The tortoise gave a sly smile. “Can you perform infinitely many actions in a finite time, Achilles?”
The athlete looked puzzled. “What’s that got to do with it?”
“Can anyone perform infinitely many actions in a finite time?” the tortoise persisted, raising the intellectual stakes.
“Doubt it,” Achilles admitted. Belatedly he added: “Whose authority?”
“Zeno of Elea.”
It was Achilles’s turn to look worried. “I’ve heard of him. Some sort of high-powered philosopher-cum-mathematician, right?”
The tortoise nodded. “Yup, that’s the guy.”
Achilles tried to seize the initiative. “Some say he’s a bit of a sophist.”
“Sorry?”
“Nutcase, basically. Puts forward arguments so slippery they’re impossible to grasp. Claims to prove motion is impossible, arrows can’t fly... Which is mad.”
To emphasise the point Achilles walked over to where he’d stashed his bow and quiver, picked out an arrow, and shot it past the finish line of the race course.
“That’s just empirical evidence,” the tortoise sneered. “A blatant appeal to the senses. No logical content whatsoever.”
Achilles stared at him. “So, Mr hoity-toitoisy logician, what’s your logical content?”
“I get 100 cubits start, right?”
“That’s what we agreed.”
“So you reach that point... when?”
“After ten seconds.”
“Have you caught me by then?”
“Near enough.”
“No! Not near enough. In ten seconds I’ve moved on another cubit.”
“Sure. But it will take me only another tenth a second to get to that point.”
“Agreed. Will you have caught me by then?”
“I’ll be a lot closer.”
“Close, but no banana. I’ll have moved on a hundredth of a cubit. Still in front. Hmm, that’s a good marketing slogan. ‘The one in front is a Tortoise.’ ”
“I can cover that in one ten thousandth of a second.”
“Of course you can. I’m sure you will. But by then—”
Achilles puffed out his cheeks. “Oh, Hades. You’ll still be in front.”
The tortoise nodded. “You do realise we can carry on like this indefinitely, don’t you? What that clever Mr. Aristotle calls ‘potential infinity’?”
Achilles’s face was a picture. “This is a trick, right?”
“A paradox,” the tortoise said. “In order to catch me, let alone pass me, you have to carry out infinitely many actions. You just admitted no one can do that.”
Achilles mulled that over. “Hang on,” he said. “Didn’t Zeno propose some other paradoxes?”
“He... did,” the tortoise said warily.
“There’s one called... the dichotomy? I’m sure that was it.”
“The dichotomy,” said the tortoise, “is completely irrel—”
“Not so quick, little chelonian nuisance.”
“I don’t do quick,” said the tortoise. “That’s the point—"
“Stop trying to change the subject. ‘That which is in motion must arrive at the half-way mark before it arrives at the goal’, right?”
“Yes, but—”
“So before you can get anywhere at all, you have to first get half way, and before that, you have to get a quarter of the way, and before that... I can keep halving the distance indefinitely! So you have to carry out infinitely many tasks even to get started!”
There was a long silence.
“The same applies to you,” the tortoise objected. “I’ll stay in front forever.”
“You’re just clutching at straws,” Achilles declared. He picked the tortoise up, walked 100 cubits, and plonked it on the sand. Then walked back to the start line. “On your marks, get set — go!”
After 10 10/99 seconds, Achilles found himself level with the tortoise. As he shot past he shouted over his shoulder: “Are we half way there yet?”
The only reply was a dignified silence.
Eventually, the tortoise, flagging, carapaced the finish line and collapsed in a heap, not that anyone would notice. Achilles, sitting cross-legged just past the line, not even having worked up a sweat, had spent more than two hours watching him struggle home.
“I won!” Achilles said. “Told you so.”
“I still maintain,” the tortoise croaked between deep breaths, “that your claim is based solely on empirical evidence. Crude observation rather than rational deduction.”
“Pretty convincing evidence, though,” Achilles pointed out. “Something must be wrong with Zeno’s proof.”
“What?”
“No idea. Probably all that guff about infinity. Infinity is weird.”
The tortoise nodded. “On that, my friend, we can agree.”
Ian Stewart FRS CMath FIMA is a British mathematician, who is best-known for his work on bifurcation theory, pattern formation and biomathematics, and catastrophe theory. As an Emeritus Professor at the University of Warwick, Ian splits his time between mathematics research and work to increase awareness of mathematics and science. He is best known for his popular science writing and is also a critically acclaimed science-fiction author. His awards include the Royal Society’s Faraday Medal (1995), the IMA Gold Medal (2000) and the AAAS Public Understanding of Science and Technology Award (2001).
October 19, 2023
